Regarding the tightness of the linear approximation technique, there is a plenty of works focusing on narrowing the input-output zone of non-linear functions in the network to gain larger certified robustness bounds [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. Some works propose tight linear bounds of non-linear functions by producing the minimal over-approximation zone, resulting in much more precise certified robustness bounds. Zhang et al. [7]} define the linear bounds producing the minimal over-approximation area as Neuron-wise Tightest linear bounds. Concretely, POPQORN [4]}, CNN-Cert [5]}, and VeriNet [10]} propose Neuron-wise Tightest robustness bounds of \(\sigma (x)tanh(y),x\cdot \sigma (y)\)  functions, ReLU function, and Sigmoid/Tanh/Arctan function, respectively.
Recently, Zhang et al. [7]} introduce the notion of Network-wise Tightest and propose Network-wise Tightest linear bounds for monotonous CNNs with Sigmoid/Tanh/Arctan function.
Unfortunately, regarding the approximation methods' performances on CNNs with Sigmoid/Tanh/Arctan function, Neuron-wise Tightest technique(VeriNet) is limited in precision while Network-wise Tightest technique(Newise [7]}) is limited in scalability. Moreover, Maxpooling function is far more complex to verified as it is a multivariate function. Some attempts have been made to verify Maxpooling function but result in loose certified robustness bounds [5]}, [6]}.
